Plan:

Introduction

• 1 Sum of the first n terms of the series
  • 1.1 Some partial sum values
  • 1.2 Euler's formula
  • 1.3 Number-theoretic properties of partial sums
• 2 Convergence of series
  • 2.1 Oresme's proof
  • 2.2 Alternative proof of divergence
• 3 Partial sums
• 4 Linked rows
  • 4.1 Dirichlet series
  • 4.2 Alternating series
  • 4.3 Random harmonic series
  • 4.4 “Thinned” harmonic series
Notes

Introduction

In mathematics, a harmonic series is a sum made up of an infinite number of terms, the reciprocals of successive numbers of the natural series:

.

The series is named harmonic, since each of its terms, starting from the second, is the harmonic average of two neighboring ones.

1. Sum of the first n terms of the series

Individual members of the series tend to zero, but their sum diverges. The nth partial sum s n of a harmonic series is the nth harmonic number:

1.1. Some partial sum values

1.2. Euler's formula

In 1740, L. Euler obtained an asymptotic expression for the sum of the first n terms of the series:

,

where is the Euler-Mascheroni constant, and ln is the natural logarithm.

When the value is , therefore, for large n:

- Euler's formula for the sum of the first n terms of the harmonic series.

1.3. Number-theoretic properties of partial sums

2. Convergence of the series

at

The harmonic series diverges very slowly (in order for the partial sum to exceed 100, about 10 43 elements of the series are needed).

The divergence of the harmonic series can be demonstrated by comparing it with the telescopic series:

,

the partial sum of which is obviously equal to:

.

2.1. Oresme's proof

The proof of divergence can be constructed by grouping the terms as follows:

The last row obviously diverges. This proof comes from the medieval scientist Nicholas Orem (c. 1350).

2.2. Alternative proof of divergence

Suppose that the harmonic series converges to the sum:

Then, rearranging the fractions, we get:

Let's take it out of the second bracket:

Replace the second bracket with:

Let's move it to the left side:

Let's substitute back the sum of the series:

This equation is obviously false, since one is greater than one-half, one-third is greater than one-fourth, and so on. Thus, our assumption about the convergence of the series is wrong, and the series diverges.

not equal to 0, because each of the brackets is positive.

This means that S is infinity and our operations of adding or subtracting it from both sides of the equality are unacceptable.

3. Partial amounts

n th partial sum of the harmonic series,

called n-th harmonic number.

Difference between n th harmonic number and natural logarithm n converges to the Euler-Mascheroni constant.

The difference between different harmonic numbers is never equal to a whole number and no harmonic number except H 1 = 1 is not an integer.

4. Linked rows

4.1. Dirichlet series

A generalized harmonic series (or Dirichlet series) is a series

.

The generalized harmonic series diverges for α≤1 and converges for α>1.

The sum of the generalized harmonic series of order α is equal to the value of the Riemann zeta function:

For even numbers, this value is clearly expressed through the number pi, for example, , and already for α=3 its value is analytically unknown.

4.2. Alternating series

The first 14 partial sums of the alternating harmonic series (black segments), showing convergence to the natural logarithm of 2 (red line).

Unlike the harmonic series, in which all terms are taken with a “+” sign, the series

converges according to Leibniz's criterion. Therefore they say that such a series has conditional convergence. Its sum is equal to the natural logarithm of 2:

This formula is a special case of the Mercator Series ( English), Taylor series for the natural logarithm.

A similar series can be obtained from the Taylor series for the arctangent:

This is known as the Leibniz series.

4.3. Random harmonic series

Biron Shmuland from the University of Alberta examined the properties of a random series

Where s n independent, identically distributed random variables that take the values ​​+1 and −1 with the same probability of ½. It is shown that this sum has probability 1, and the sum of the series is a random variable with interesting properties. For example, the probability density function calculated at points +2 or −2 has a value of 0.124 999 999 999 999 999 999 999 999 999 999 999 999 999 7642 ..., differing from by less than 10 −42. Shmuland's paper explains why this value is close to, but not equal to, 1/8.

4.4. “Thinned” harmonic series

Kempner series ( English)

If we consider a harmonic series in which only terms are left whose denominators do not contain the number 9, then it turns out that the remaining sum converges to the number<80. , точнее - к 22,92067 66192 64150 34816. Более того, доказано, что если оставить слагаемые, не содержащие любой заранее выбранной последовательности цифр, то полученный ряд будет сходиться. Однако из этого будет ошибочно заключать о сходимости исходного гармонического ряда, т.к. с ростом разрядов в числе n, все меньше слагаемых берется для суммы "истонченного" ряда. Т.е. в конечном счете мы отбрасываем подавляющее большинство членов образующих сумму гармонического ряда, чтобы не превзойти ограничивающую сверху геометрическую прогрессию.

1.1. Number series and its sum

Definition 1. Let a number sequence be given. Let's form an expression

(1)

which is called number series. Numbers are called members of a number, and the expression
 common member row .

Example 1. Find the common term of the series
.

at
,

at

It is easy to see that the common term of the series .

Therefore, the required series can be written as follows

.

Let us construct a sequence from the terms of series (1) in this way :

;

;

;

Each member of this sequence represents the sum of the corresponding number of the first members of the number series.

Definition 2. Sum of first P members of series (1) is called n -th partial amount number series .

Definition 3. Number series called convergent, If
, where the number called sum of the series, and write
. If

the limit of partial sums is infinite or does not exist, then the series is called divergent.

Example 2. Check series for convergence
.

In order to calculate n-th partial amount let's imagine a common term
series in the form of a sum of simple fractions

Comparing coefficients at the same degrees n, we obtain a system of linear algebraic equations for unknown coefficients A And IN

From here we find that
, A
.

Therefore, the general term of the series has the form

Then the partial amount can be represented in the form

After opening the brackets and bringing similar terms, it will take the form

.

Let's calculate the sum of the series

Since the limit is equal to a finite number, this series converges .

Example 2. Check series for convergence

- infinite geometric progression.

As is known, the sum of the first P members of the geometric progression at q 1 is equal
.

Then we have the following cases :

1. If
, That

2. If
, That
, i.e. the row diverges.

3. If
, then the series has to be seen then
, i.e. the row diverges.

4. If
, then the series has to be seen then
, if the partial sum has an even number of terms and
, if the number is odd, i.e.
does not exist, therefore the series diverges.

Definition 4. Difference between the sum of the series S and partial amount called the rest of the series and is designated
, i.e.
.

Since for convergent series
, That
,

those. will be b.m.v. at
. So the value is an approximate value of the sum of the series.

From the definition of the sum of a series, the properties of convergent series follow:

1. If the rows And converge, i.e. have corresponding amounts S And Q, then the series converges, where
, and its sum is equal A S + B Q.

2. If the series converges , then the series obtained from this converges

series by dropping or adding a finite number of terms. The opposite is also true.

1.2. A necessary sign of convergence. Harmonic series

Theorem. If the row converges, then the common term of the series tends to zero as
, i.e.
.

Indeed, we have

Then , which was what needed to be proven.

Consequence. If
, then the series diverges . The converse, generally speaking, is not true, as will be shown below.

Definition 5. View series called harmonic.

For this series, the necessary characteristic is satisfied, since
.

At the same time, it is divergent. Let's show it

Thus, the harmonic series diverges.

Topic 2 : Sufficient signs of series convergence

with positive terms

2.1. Signs of comparison

Let two series with positive terms be given:

Sign of comparison. If for all members of series (1) and (2), starting from a certain number, the inequality
and series (2) converges, then series (1) also converges. Likewise, if
and series (2) diverges, then series (1) also diverges.

Let And respectively, partial sums of rows (1-2), and Q sum of series (2). Then for large enough P we have

Because
and limited, then
, i.e. series (1) converges.

The second part of the sign is proved in a similar way.

Example 3. Examine the series for convergence

.

Let's compare with the members of the series
.

Beginning with
, we have
.

Since the series converges
, then this series also converges.

In practice, it is often more convenient to use the so-called limiting criterion for comparison, which follows from the previous one.

Limit of comparison. If for two series (1-2) with positive terms the condition is met

, That

from the convergence of series (1) follows the convergence of series (2), and from the divergence of series (1) follows the divergence of series (2) , those. the rows behave the same.

Example 4. Examine the series for convergence
.

As a series for comparison, let’s take the harmonic series,

which is divergent.

and, therefore, our series diverges.

Comment. It is often convenient to use the so-called generalized harmonic row , which, as will be shown below, converges at
and diverges at
.

Harmonic series- a sum made up of an infinite number of terms, inverses of successive numbers of the natural series:

texvc not found; See math/README - help with setup.): \sum_(k=1)^\mathcal(\infty) \frac(1)(k)=1 + \frac(1)(2) + \frac(1 )(3) + \frac(1)(4) + \cdots +\frac(1)(k) + \cdots .

Sum of the first n terms of the series

Individual members of the series tend to zero, but their sum diverges. The nth partial sum s n of a harmonic series is the nth harmonic number:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): s_n=\sum_(k=1)^n \frac(1)(k)=1 + \frac(1)(2) + \frac(1)(3 ) + \frac(1)(4) + \cdots +\frac(1)(n)

Some partial sum values

Euler's formula

At Unable to parse expression (Executable file texvc meaning Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \varepsilon _n \rightarrow 0, therefore, for large Unable to parse expression (Executable file texvc :

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): s_n\approx \ln(n) + \gamma- Euler's formula for the sum of the first Unable to parse expression (Executable file texvc not found; See math/README for setup help.): n members of the harmonic series.

A more accurate asymptotic formula for the partial sum of the harmonic series:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): s_n \asymp \ln(n) + \gamma + \frac(1)(2n) - \frac(1)(12n^2) + \frac(1)( 120n^4) - \frac(1)(252n^6) \dots = \ln(n) + \gamma + \frac(1)(2n) - \sum_(k=1)^(\infty) \frac (B_(2k))(2k\,n^(2k)), Where Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): B_(2k)- Bernoulli numbers.

This series diverges, but the error in its calculations never exceeds half of the first discarded term.

Number-theoretic properties of partial sums

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \forall n>1\;\;\;\;s_n\notin\mathbb(N)

Divergence of series

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): s_n\rightarrow \infty at Unable to parse expression (Executable file texvc not found; See math/README for setup help.): n\rightarrow \infty

The harmonic series diverges very slowly (for the partial sum to exceed 100, about 10 43 elements of the series are needed).

The divergence of the harmonic series can be demonstrated by comparing it with the telescopic series:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): v_n = \ln(n+1)-\ln n = \ln \left(1+\frac(1)(n)\right)\underset(+\infty )(\sim)\frac (1)(n) ,

the partial sum of which is obviously equal to:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_(i=1)^(n-1) v_i= \ln n \sim H_n .

Oresme's proof

The proof of divergence can be constructed by grouping the terms as follows:

Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \begin(align) \sum_(k=1)^\infty \frac(1)(k) & () = 1 + \left[\frac(1)( 2)\right] + \left[\frac(1)(3) + \frac(1)(4)\right] + \left[\frac(1)(5) + \frac(1)(6) + \frac(1)(7) + \frac(1)(8)\right] + \left[\frac(1)(9)+\cdots\right] +\cdots \\ & () > 1 + \left[\frac(1)(2)\right] + \left[\frac(1)(4) + \frac(1)(4)\right] + \left[\frac(1)(8) + \frac(1)(8) + \frac(1)(8) + \frac(1)(8)\right] + \left[\frac(1)(16)+\cdots\right] +\ cdots \\ & () = 1 + \ \frac(1)(2)\ \ \ + \quad \frac(1)(2) \ \quad + \ \qquad\quad\frac(1)(2)\ qquad\ \quad \ + \quad \ \ \frac(1)(2) \ \quad + \ \cdots. \end(align)

The last row obviously diverges. This proof comes from the medieval scientist Nicholas Orem (c. 1350).

Alternative proof of divergence

Difference between Unable to parse expression (Executable file texvc not found; See math/README for setup help.): n th harmonic number and natural logarithm Unable to parse expression (Executable file texvc not found; See math/README for setup help.): n converges to the Euler–Mascheroni constant.

The difference between different harmonic numbers is never equal to a whole number and no harmonic number except Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): H_1=1, is not an integer.

Related series

Dirichlet series

A generalized harmonic series (or Dirichlet series) is a series

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_(k=1)^\infty \frac(1)(k^\alpha)=1 + \frac(1)(2^\alpha) + \frac (1)(3^\alpha) + \frac(1)(4^\alpha) + \cdots +\frac(1)(k^\alpha) + \cdots .

The generalized harmonic series diverges at Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \alpha \leqslant 1 and converges at Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \alpha > 1 .

Sum of generalized harmonic series of order Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \alpha equal to the value of the Riemann zeta function:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_(k=1)^\mathcal(1) \frac(1)(k^\alpha)=\zeta(\alpha)

For even numbers, this value is explicitly expressed in terms of pi, for example, Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \zeta(2)=\frac(\pi^2)(6), and already for α=3 its value is analytically unknown.

Another illustration of the divergence of the harmonic series can be the relation Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \zeta(1+\frac(1)(n)) \sim n .

Alternating series

Unlike the harmonic series, in which all terms are taken with a “+” sign, the series

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_(n = 1)^\infty \frac((-1)^(n + 1))(n) \;=\; 1 \,-\, \frac(1)(2) \,+\, \frac(1)(3) \,-\, \frac(1)(4) \,+\, \frac(1) (5) \,-\, \cdots Unable to parse expression (Executable file texvc not found; See math/README for setup help.): 1 \,-\, \frac(1)(2) \,+\, \frac(1)(3) \,-\, \frac(1)( 4) \,+\, \frac(1)(5) \,-\, \cdots \;=\; \ln 2.

This formula is a special case of the Mercator series ( English), Taylor series for the natural logarithm.

A similar series can be derived from the Taylor series for the arctangent:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_(n = 0)^\infty \frac((-1)^(n))(2n+1) \;\;=\;\; 1 \,-\, \frac(1)(3) \,+\, \frac(1)(5) \,-\, \frac(1)(7) \,+\, \cdots \;\ ;=\;\; \frac(\pi)(4).

This relationship is known as the Leibniz series.

Random harmonic series

In 2003, the properties of a random series were studied

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum_(n=1)^(\infty)\frac(s_(n))(n),

Where Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): s_n- independent, identically distributed random variables that take values ​​+1 and −1 with the same probability of ½. It is shown that this series converges with probability 1, and the sum of the series is a random variable with interesting properties. For example, the probability density function calculated at points +2 or −2 has the value:

0,124 999 999 999 999 999 999 999 999 999 999 999 999 999 7 642 …,

differing from ⅛ by less than 10 −42.

“Thinned” harmonic series

Kempner series ( English)

If we consider a harmonic series in which only terms are left whose denominators do not contain the number 9, then it turns out that the remaining sum converges to the number<80 . Более того, доказано, что если оставить слагаемые, не содержащие любой заранее выбранной последовательности цифр, то полученный ряд будет сходиться. Однако из этого будет ошибочно заключать о сходимости исходного гармонического ряда, так как с ростом разрядов в числе Unable to parse expression (Executable file texvc not found; See math/README for setup help.): n, fewer and fewer terms are taken for the sum of the “thinned” series. That is, ultimately, the overwhelming majority of terms forming the sum of the harmonic series are discarded so as not to exceed the geometric progression limiting from above.

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Notes

View this template Sequences and series Sequences Rows, main Number series (actions with number series) Functional series Other types of rows Signs of convergence View this template Signs of series convergence For all rows Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum^\infty_(n=1)a_n For positive signs rows For those taking turns rows For rows of the form Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \sum^\infty_(n=1)a_n b_n For function rows For Fourier series

An excerpt characterizing the Harmonic series

The terrible day was coming to an end. I sat by the open window, feeling and hearing nothing. The world became frozen and joyless for me. It seemed that he existed separately, not making his way into my tired brain and not touching me in any way... On the windowsill, playing, the restless “Roman” sparrows were still squealing. Below there were human voices and the usual daytime noise of a bustling city. But all this came to me through some very dense “wall”, which almost did not allow sounds to pass through... My usual inner world was empty and deaf. He became completely alien and dark... The sweet, affectionate father no longer existed. He followed Girolamo...
But I still had Anna. And I knew that I had to live in order to save at least her from a sophisticated killer who called himself the “vicar of God,” the Holy Pope... It was hard to even imagine, if Caraffa was just his “viceroy,” then what kind of beast must turn out to be this beloved God of his?!. I tried to get out of my “frozen” state, but as it turned out, it was not so easy - the body did not obey at all, not wanting to come to life, and the tired Soul was looking only for peace... Then, seeing that nothing good was working out, I just decided to leave myself alone, letting everything take its course.
Without thinking anything else, and without deciding anything, I simply “flew away” to where my wounded Soul was striving, in order to be saved... To rest and forget at least a little, going far from the evil “earthly” world to where only light reigned ...
I knew that Caraffa would not leave me alone for long, despite what I had just gone through, on the contrary - he would consider that the pain had weakened and disarmed me, and perhaps at this moment he would try to force me to surrender by inflicting some kind of - another terrifying blow...
The days passed. But, to my greatest surprise, Caraffa did not appear... This was a huge relief, but, unfortunately, it did not allow me to relax. Because every moment I expected what new meanness his dark, evil soul would come up with for me...
The pain gradually dulled every day, mainly thanks to an unexpected and joyful incident that happened a couple of weeks ago and completely stunned me - I had the opportunity to hear my deceased father!..
I couldn’t see him, but I heard and understood every word very clearly, as if my father was next to me. At first I didn’t believe it, thinking that I was just delirious from complete exhaustion. But the call was repeated... It was indeed the father.
With joy, I could not come to my senses and was still afraid that suddenly, right now, he would just up and disappear!.. But my father did not disappear. And having calmed down a little, I was finally able to answer him...
– Is it really you!? Where are you now?.. Why can’t I see you?
– My daughter... You don’t see because you’re completely exhausted, dear. Anna sees that I was with her. And you will see, dear. You just need time to calm down.
Pure, familiar warmth spread throughout my entire body, enveloping me in joy and light...
- How are you, father!? Tell me what it looks like, this other life?.. What is it like?
– She’s wonderful, dear!.. Only she’s still unusual. And so different from our former earthly one!.. Here people live in their own worlds. And they are so beautiful, these “worlds”!.. But I still can’t do it. Apparently, it’s still too early for me... – the voice fell silent for a second, as if deciding whether to speak further.
- Your Girolamo met me, daughter... He is as alive and loving as he was on Earth... He misses you very much and yearns. And he asked me to tell you that he loves you just as much there... And is waiting for you whenever you come... And your mother is with us too. We all love and are waiting for you, dear. We really miss you... Take care of yourself, daughter. Don’t let Karaffa have the joy of mocking you.
– Will you come to me again, father? Will I hear you again? – afraid that he would suddenly disappear, I prayed.
- Calm down, daughter. Now this is my world. And Caraffa's power does not extend to him. I will never leave you or Anna. I will come to you whenever you call. Calm down, dear.
- How do you feel, father? Do you feel anything?.. – a little embarrassed by my naive question, I still asked.
– I feel everything that I felt on Earth, only much brighter. Imagine a pencil drawing that is suddenly filled with colors - all my feelings, all my thoughts are much stronger and more colorful. And one more thing... The feeling of freedom is amazing!.. It seems that I am the same as I have always been, but at the same time completely different... I don’t know how to explain it to you more precisely, dear... As if I can immediately embrace everything the world, or just fly far, far, to the stars... Everything seems possible, as if I can do anything I want! It’s very difficult to tell, to put into words... But believe me, daughter, it’s wonderful! And one more thing... I now remember all my lives! I remember everything that once happened to me... It’s all amazing. This “other” life, as it turned out, is not so bad... Therefore, don’t be afraid, daughter, if you have to come here, we will all be waiting for you.
– Tell me, father... Is there really a wonderful life waiting for people like Caraffa there too?.. But, in that case, this is again a terrible injustice!.. Will everything really be like on Earth again?!.. Is he really will never receive retribution?!!
- Oh no, my joy, there is no place for Karaffa here. I've heard people like him go into a terrible world, but I haven't been there yet. They say this is what they deserve!.. I wanted to see it, but I haven’t had time yet. Don't worry, daughter, he'll get what he deserves when he gets here.
“Can you help me from there, father?” I asked with hidden hope.
– I don’t know, dear... I haven’t understood this world yet. I am like a child taking its first steps... I have to first “learn to walk” before I can answer you... And now I have to go. Sorry, honey. First I must learn to live between our two worlds. And then I will come to you more often. Take courage, Isidora, and never give in to Karaffa. He will definitely get what he deserves, believe me.
My father’s voice became quieter until it became completely thin and disappeared... My soul calmed down. It really was HIM!.. And he lived again, only now in his own, still unfamiliar to me, posthumous world... But he still thought and felt, as he himself had just said - even much brighter than when he lived on Earth. I could no longer be afraid that I would never know about him... That he had left me forever.
But my feminine soul, in spite of everything, still grieved for him... About the fact that I couldn’t just hug him like a human being when I felt lonely... That I couldn’t hide my melancholy and fear on his wide chest, wanting peace... That his strong, gentle palm could no longer stroke my tired head, as if saying that everything would work out and everything would definitely be fine... I desperately missed these small and seemingly insignificant, but such dear, purely “human” joys, and the soul was hungry for them, unable to find peace. Yes, I was a warrior... But I was also a woman. His only daughter, who used to always know that even if the worst happened, my father would always be there, always be with me... And I painfully missed all this...
Somehow shaking off the surging sadness, I forced myself to think about Karaffa. Such thoughts immediately sobered me up and forced me to gather myself internally, since I perfectly understood that this “peace” was just a temporary respite...
But to my greatest surprise, Caraffa still did not appear...
Days passed and anxiety grew. I tried to come up with some explanation for his absence, but, unfortunately, nothing serious came to mind... I felt that he was preparing something, but I couldn’t guess what. Exhausted nerves gave way. And in order not to completely go crazy from waiting, I started walking around the palace every day. I was not forbidden to go out, but it was also not approved, therefore, not wanting to continue being locked up, I decided for myself that I would go for a walk... despite the fact that perhaps someone would not like it. The palace turned out to be huge and unusually rich. The beauty of the rooms amazed the imagination, but personally I could never live in such eye-catching luxury... The gilding of the walls and ceilings was oppressive, infringing on the craftsmanship of the amazing frescoes, suffocating in the sparkling environment of golden tones. I paid tribute with pleasure to the talent of the artists who painted this wonderful home, admiring their creations for hours and sincerely admiring the finest craftsmanship. So far no one has bothered me, no one has ever stopped me. Although there were always some people who, having met, bowed respectfully and moved on, each rushing about his own business. Despite such false “freedom,” all this was alarming, and each new day brought more and more anxiety. This “calm” could not last forever. And I was almost sure that it would definitely “give birth” to some terrible and painful misfortune for me...

A necessary criterion for the convergence of series (prove).

Theorem 1.(a necessary condition for the convergence of a number series). If the number series converges, That .

Proof. The series converges, i.e. there is a limit. Notice, that .

Let's consider. Then . From here, .

Corollary 1.If the condition is not met, then the series diverges.

Note 1. The condition is not sufficient for the convergence of a number series. For example, harmonic series diverges, although it does occur.

Definition 1. Number series a n +1 +a n+2 +…=, obtained from a given row by discarding the first P members is called n- m the remainder of this row and is designated Rn.

Theorem 2.If the number series converges, then any remainder converges. Back:If at least one remainder of the series converges, then the series itself converges. Moreover, for any n ON the equality S=S n+Rn .

Corollary 2. The convergence or divergence of a number series will not change if you remove or add the first few terms.

Corollary 3..

32. Comparison criteria and sign for positive series

Theorem 1(a sign of comparing series with positive terms in inequalities) . LetAnd - series with non-negative terms, and for each n ON condition a n is satisfied£ bn. Then:

1) from the convergence of the serieswith large terms the series convergeswith smaller members;

2) from the divergence of the serieswith smaller terms the series divergeswith big dicks.

Note 1. The theorem is true if the condition and n£ b n executed from some number NÎ N .

Theorem 2(a sign of comparison of series with positive terms in limit form) .

LetAnd - series with non-negative terms and there is . Then these series converge or diverge simultaneously .

33. D'Alembert's test for convergence of positive-sign series

Theorem 1(D'Alembert's sign). Let - a series with positive terms exists .

Then the series converges at q<1 and diverges at q>1 .

Proof. Let q<1. Зафиксируем число R such that q<p< 1. По определению limit of the number sequence, from some number NÎ N inequality holds a n +1 /a n<p, those. a n +1 <p×a n . Then a N +1 < p×a N , a N +2 <p 2 ×a N . It is easy to show by induction that for any kÎ N inequality true , a N + k<p k ×a N . But the series converges like a geometric series ( p<1). Следовательно, по признаку сравнения рядов с неотрицательными членами, ряд also converges. Consequently, the series also converges (by Theorem 2.2).

Let q>1. Then from some number NÎ N inequality true a n +1 /a n>1, i.e. a n +1 >a n. Therefore, from the number N subsequence ( a n) is increasing and the condition is not satisfied. From here, by Corollary 2.1, it follows that the series diverges at q>1.

Note 1. Using the integral test, it is easy to check that the number series converges if A>1, and diverges if a£1. Row called harmonic series, and the series with arbitrary aÎ R called generalized harmonic series.

34. Alternating rows. Leibniz test for the convergence of sign of alternating series

The study of series with terms of arbitrary signs is a more difficult task, but in two cases there are convenient signs: for series of alternating signs - Leibniz's theorem; For absolutely convergent series, we apply any sign of studying series with non-negative terms.

Definition 1. The number series is called signalternating, if any two adjacent terms have opposite signs, i.e. the series has the form or , where a n>0 for each nÎ N .

Theorem 1(Leibniz). An alternating series converges if:

1) (a n) - non-increasing sequence;

2) at.

In this case, the modulus of the sum of the alternating series does not exceed the modulus of its first term, i.e.|S|£ a 1 .

Rows for teapots. Examples of solutions

I welcome all survivors to the second year! In this lesson, or rather, in a series of lessons, we will learn how to manage rows. The topic is not very complicated, but mastering it will require knowledge from the first year, in particular, you need to understand what is a limit, and be able to find the simplest limits. However, it’s okay, as I explain, I will provide relevant links to the necessary lessons. To some readers, the topic of mathematical series, solution methods, signs, theorems may seem peculiar, and even pretentious, absurd. In this case, you don’t need to be too “loaded”; we accept the facts as they are and simply learn to solve typical, common tasks.

1) Rows for dummies, and for samovars immediately content :)

For super-fast preparation on the topic There is an express course in pdf format, with the help of which you can really “raise” your practice literally in a day.

The concept of a number series

In general number series can be written like this: .
Here:
– mathematical sum icon;
– common term of the series(remember this simple term);
– “counter” variable. The notation means that summation is carried out from 1 to “plus infinity”, that is, first we have , then , then , and so on - to infinity. Instead of a variable, a variable or is sometimes used. Summation does not necessarily start from one; in some cases it can start from zero, from two, or from any natural number.

In accordance with the “counter” variable, any series can be expanded:
- and so on, ad infinitum.

Components - This NUMBERS which are called members row. If they are all non-negative (greater than or equal to zero), then such a series is called positive number series.

Example 1

This, by the way, is already a “combat” task - in practice, quite often it is necessary to write down several terms of a series.

First, then:
Then, then:
Then, then:

The process can be continued indefinitely, but according to the condition it was required to write the first three terms of the series, so we write down the answer:

Please note the fundamental difference from number sequence,
in which the terms are not summed up, but are considered as such.

Example 2

Write down the first three terms of the series

This is an example for you to solve on your own, the answer is at the end of the lesson

Even for a series that is complex at first glance, it is not difficult to describe it in expanded form:

Example 3

Write down the first three terms of the series

In fact, the task is performed orally: mentally substitute into the common term of the series first, then and. Eventually:

We leave the answer as follows: It is better not to simplify the resulting series terms, that is do not perform actions: , , . Why? The answer is in the form it is much easier and more convenient for the teacher to check.

Sometimes the opposite task occurs

Example 4

There is no clear solution algorithm here, you just need to see the pattern.
In this case:

To check, the resulting series can be “written back” in expanded form.

Here's an example that's a little more complicated to solve on your own:

Example 5

Write down the sum in collapsed form with the common term of the series

Perform a check by again writing the series in expanded form

Convergence of number series

One of the key objectives of the topic is study of series for convergence. In this case, two cases are possible:

1) Rowdiverges. This means that an infinite sum is equal to infinity: or sums in general does not exist, as, for example, in the series
(here, by the way, is an example of a series with negative terms). A good example of a divergent number series was found at the beginning of the lesson: . Here it is quite obvious that each next member of the series is greater than the previous one, therefore and, therefore, the series diverges. An even more trivial example: .

2) Rowconverges. This means that an infinite sum is equal to some finite number: . Please: – this series converges and its sum is zero. As a more meaningful example, we can cite infinitely decreasing geometric progression, known to us since school: . The sum of the terms of an infinitely decreasing geometric progression is calculated using the formula: , where is the first term of the progression, and is its base, which is usually written in the form correct fractions In this case: , . Thus: A finite number is obtained, which means the series converges, which is what needed to be proved.

However, in the vast majority of cases find the sum of the series is not so simple, and therefore in practice, to study the convergence of a series, special signs that have been proven theoretically are used.

There are several signs of series convergence: necessary test for the convergence of a series, comparison tests, D'Alembert's test, Cauchy's tests, Leibniz's sign and some other signs. When to use which sign? It depends on the common member of the series, figuratively speaking, on the “filling” of the series. And very soon we will sort everything out.

! To further learn the lesson, you must understand well what is a limit and it is good to be able to reveal the uncertainty of a type. To review or study the material, please refer to the article Limits. Examples of solutions.

A necessary sign of convergence of a series

If a series converges, then its common term tends to zero: .

The converse is not true in the general case, i.e., if , then the series can either converge or diverge. And therefore this sign is used to justify divergences row:

If the common term of the series does not tend to zero, then the series diverges

Or in short: if , then the series diverges. In particular, a situation is possible where the limit does not exist at all, as, for example, limit. So they immediately justified the divergence of one series :)

But much more often, the limit of a divergent series is equal to infinity, and instead of “x” it acts as a “dynamic” variable. Let's refresh our knowledge: limits with “x” are called limits of functions, and limits with the variable “en” are called limits of numerical sequences. The obvious difference is that the variable "en" takes discrete (discontinuous) natural values: 1, 2, 3, etc. But this fact has little effect on methods for solving limits and methods for disclosing uncertainties.

Let us prove that the series from the first example diverges.
Common member of the series:

Conclusion: row diverges

The necessary feature is often used in real practical tasks:

Example 6

We have polynomials in the numerator and denominator. The one who carefully read and comprehended the method of disclosing uncertainty in the article Limits. Examples of solutions, I probably caught that when the highest powers of the numerator and denominator equal, then the limit is finite number .

Divide the numerator and denominator by

Series under study diverges, since the necessary criterion for the convergence of the series is not fulfilled.

Example 7

Examine the series for convergence

This is an example for you to solve on your own. Full solution and answer at the end of the lesson

So, when we are given ANY number series, Firstly we check (mentally or on a draft): does its common term tend to zero? If it doesn’t, we formulate a solution based on examples No. 6, 7 and give an answer that the series diverges.

What types of apparently divergent series have we considered? It is immediately clear that series like or diverge. The series from examples No. 6, 7 also diverge: when the numerator and denominator contain polynomials, and the leading power of the numerator is greater than or equal to the leading power of the denominator. In all these cases, when solving and preparing examples, we use the necessary sign of convergence of the series.

Why is the sign called necessary? Understand in the most natural way: in order for a series to converge, necessary, so that its common term tends to zero. And everything would be great, but there’s more not enough. In other words, if the common term of a series tends to zero, THIS DOES NOT MEAN that the series converges– it can both converge and diverge!

Meet:

This series is called harmonic series. Please remember! Among the number series, he is a prima ballerina. More precisely, a ballerina =)

It's easy to see that , BUT. In the theory of mathematical analysis it has been proven that harmonic series diverges.

You should also remember the concept of a generalized harmonic series:

1) This row diverges at . For example, the series , , diverge.
2) This row converges at . For example, the series , , , converge. I emphasize once again that in almost all practical tasks it is not at all important to us what the sum of, for example, the series is equal to, the very fact of its convergence is important.

These are elementary facts from the theory of series that have already been proven, and when solving any practical example, you can safely refer, for example, to the divergence of a series or the convergence of a series.

In general, the material in question is very similar to study of improper integrals, and it will be easier for those who have studied this topic. Well, for those who haven’t studied it, it’s doubly easier :)

So, what to do if the common term of the series TENDS to zero? In such cases, to solve examples you need to use others, sufficient signs of convergence/divergence:

Comparison criteria for positive number series

I draw your attention, that here we are talking only about positive number series (with non-negative terms).

There are two signs of comparison, one of them I will simply call a sign of comparison, another - limit of comparison.

Let's first consider comparison sign, or rather, the first part of it:

Consider two positive number series and . If known, that the series – converges, and, starting from some number, the inequality is satisfied, then the series also converges.

In other words: From the convergence of the series with larger terms follows the convergence of the series with smaller terms. In practice, the inequality often holds for all values:

Example 8

Examine the series for convergence

First, let's check(mentally or in draft) execution:
, which means it was not possible to “get off with little blood.”

We look into the “pack” of the generalized harmonic series and, focusing on the highest degree, we find a similar series: It is known from theory that it converges.

For all natural numbers, the obvious inequality holds:

and larger denominators correspond to smaller fractions:
, which means, based on the comparison criterion, the series under study converges together with next to .

If you have any doubts, you can always describe the inequality in detail! Let us write down the constructed inequality for several numbers “en”:
If , then
If , then
If , then
If , then
….
and now it is absolutely clear that inequality fulfilled for all natural numbers “en”.

Let's analyze the comparison criterion and the solved example from an informal point of view. Still, why does the series converge? Here's why. If a series converges, then it has some final amount: . And since all members of the series less corresponding terms of the series, then it is clear that the sum of the series cannot be greater than the number, and even more so, cannot be equal to infinity!

Similarly, we can prove the convergence of “similar” series: , , etc.

! note, that in all cases we have “pluses” in the denominators. The presence of at least one minus can seriously complicate the use of the product in question. comparison sign. For example, if a series is compared in the same way with a convergent series (write out several inequalities for the first terms), then the condition will not be satisfied at all! Here you can dodge and select another convergent series for comparison, for example, but this will entail unnecessary reservations and other unnecessary difficulties. Therefore, to prove the convergence of a series it is much easier to use limit of comparison(see next paragraph).

Example 9

Examine the series for convergence

And in this example, I suggest you consider for yourself second part of the comparison attribute:

If known, that the series – diverges, and starting from some number (often from the very first), the inequality is satisfied, then the series also diverges.

In other words: From the divergence of a series with smaller terms follows the divergence of a series with larger terms.

What should be done?
It is necessary to compare the series under study with a divergent harmonic series. For a better understanding, construct several specific inequalities and make sure that the inequality is fair.

The solution and sample design are at the end of the lesson.

As already noted, in practice, the comparison criterion just discussed is rarely used. The real workhorse of number series is limit of comparison, and in terms of frequency of use it can only compete with d'Alembert's sign.

Limit test for comparing numerical positive series

Consider two positive number series and . If the limit of the ratio of the common terms of these series is equal to finite non-zero number: , then both series converge or diverge simultaneously.

When is the limiting criterion used? The limiting criterion for comparison is used when the “filling” of the series is polynomials. Either one polynomial in the denominator, or polynomials in both the numerator and denominator. Optionally, polynomials can be located under the roots.

Let's deal with the row for which the previous comparison sign has stalled.

Example 10

Examine the series for convergence

Let's compare this series with a convergent series. We use the limiting criterion for comparison. It is known that the series converges. If we can show that equals finite, non-zero number, it will be proven that the series also converges.

A finite non-zero number is obtained, which means the series under study is converges together with next to .

Why was the series chosen for comparison? If we had chosen any other series from the “cage” of the generalized harmonic series, then we would not have succeeded in the limit finite, non-zero numbers (you can experiment).

Note: when we use the limiting comparison criterion, doesn't matter, in what order to compose the relation of common members, in the example considered, the relation could be compiled the other way around: - this would not change the essence of the matter.